A122746 -id:A122746 - cp's OEIS frontend

A056309 Number of reversible strings with n beads using exactly two different colors.

0, 1, 4, 8, 18, 34, 70, 134, 270, 526, 1054, 2078, 4158, 8254, 16510, 32894, 65790, 131326, 262654, 524798, 1049598, 2098174, 4196350, 8390654, 16781310, 33558526, 67117054, 134225918, 268451838, 536887294, 1073774590, 2147516414, 4295032830, 8590000126

Offset: 1

Marks R. Nester

A string and its reverse are considered to be equivalent.

			For n=3, the four rows are ABA, BAB, AAB, and ABB, the last two being respectively equivalent to BAA and BBA, with which they form chiral pairs. - _Robert A. Russell_, Sep 25 2018

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Harvey P. Dale, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (3,0,-6,4).

Cf. A000079, A005418.
Equals (A000918 + A056453) / 2.
a(n) = A000918(n) - A122746(n-2) = A122746(n-2) + A056453(n).

[2^(n-1)+2^((n-1) div 2)-2: n in [1..40]]; // Vincenzo Librandi, Sep 29 2018

seq(2^(n-1) + 2^floor((n-1)/2) - 2, n=1..34); # Peter Luschny, Nov 25 2017

Rest[CoefficientList[Series[x^2(1+x-4x^2)/(1-3x+6x^3-4x^4),{x,0,30}],x]] (* or *) LinearRecurrence[{3,0,-6,4},{0,1,4,8},30] (* Harvey P. Dale, Feb 18 2012 *)

Vec(x^2*(1+x-4*x^2)/(1-3*x+6*x^3-4*x^4) + O(x^40)) \\ Colin Barker, Nov 24 2017

a(n) = 2^(n-1)+2^((n-1)\2)-2; \\ Altug Alkan, Sep 25 2018

a(n) = A005418(n+1) - 2.
G.f.: x^2*(1 + x - 4*x^2)/(1 - 3*x + 6*x^3 - 4*x^4). - Colin Barker, Feb 03 2012
a(1)=0, a(2)=1, a(3)=4, a(4)=8, a(n) = 3*a(n-1) - 6*a(n-3) + 4*a(n-4). - Harvey P. Dale, Feb 18 2012
From Colin Barker, Nov 24 2017: (Start)
a(n) = 2^(n/2-1) + 2^(n-1) - 2 for n even.
a(n) = 2^((n-1)/2) + 2^(n-1) - 2 for n odd. (End)
a(n) = A000079(n-1) + A056453(n-2). - Peter Luschny, Nov 25 2017
a(n) = k! (S2(n,k) + S2(ceiling(n/2),k)) / 2, where k=2 is the number of colors and S2 is the Stirling subset number. - Robert A. Russell, Sep 25 2018

A233411 The number of length n binary words with some prefix which contains two more 1's than 0's or two more 0's than 1's.

Original entry on oeis.org

0, 0, 2, 4, 12, 24, 56, 112, 240, 480, 992, 1984, 4032, 8064, 16256, 32512, 65280, 130560, 261632, 523264, 1047552, 2095104, 4192256, 8384512, 16773120, 33546240, 67100672, 134201344, 268419072, 536838144, 1073709056, 2147418112, 4294901760, 8589803520

Offset: 0

Geoffrey Critzer, Dec 09 2013

Also, the number of non-symmetric compositions of n+1, e.g. 4 can be written 1+3, 3+1, 1+1+2, or 2+1+1 (but not 4, 2+2, 1+2+1 or 1+1+1+1). - Henry Bottomley, Jun 27 2005
If we examine the set of all binary words with infinite length we find that the average length of the shortest prefix which satisfies the above conditions is 4.
a(n) is also the number of minimum distinguishing (2-)labelings of the path graph P_n for n > 1. - Eric W. Weisstein, Oct 16 2014
Also, the decimal representation of the diagonal from the origin to the corner of the n-th stage of growth of the two-dimensional cellular automaton defined by "Rule 62", based on the 5-celled von Neumann neighborhood, initialized with a single black (ON) cell at stage zero. - Robert Price, Apr 22 2017

			a(3) = 4 because we have: 000, 001, 110, 111.

S. Wolfram, A New Kind of Science, Wolfram Media, 2002; p. 170.

N. J. A. Sloane, On the Number of ON Cells in Cellular Automata, arXiv:1503.01168 [math.CO], 2015
Eric Weisstein's World of Mathematics, Elementary Cellular Automaton
S. Wolfram, A New Kind of Science
Wolfram Research, Wolfram Atlas of Simple Programs
Eric Weisstein's World of Mathematics, Distinguishing Number
Index entries for sequences related to cellular automata
Index to 2D 5-Neighbor Cellular Automata
Index to Elementary Cellular Automata
Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A233533.

nn=30;CoefficientList[Series[2x^2/(1-2x^2)/(1-2x),{x,0,nn}],x]
LinearRecurrence[{2,2,-4},{0,0,2},40] (* Harvey P. Dale, Sep 06 2015 *)

a(n)=2^n-2^ceil(n/2) \\ Charles R Greathouse IV, Dec 09 2013

G.f.: 2*x^2/( (1 - 2*x^2)*(1-2x) ).
a(n) = 2^n - 2^ceiling(n/2).
a(n) = 2*A032085(n) = 2*A122746(n-2) for n>=2. - Alois P. Heinz, Dec 09 2013

Misplaced comment added by Andrew Howroyd, Sep 30 2017

A320751 Array read by antidiagonals: T(n,k) is the number of chiral pairs of color patterns (set partitions) in a row of length n using k or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 0, 0, 1, 4, 6, 0, 0, 0, 1, 4, 16, 12, 0, 0, 0, 1, 4, 20, 52, 28, 0, 0, 0, 1, 4, 20, 80, 169, 56, 0, 0, 0, 1, 4, 20, 86, 336, 520, 120, 0, 0, 0, 1, 4, 20, 86, 400, 1344, 1600, 240, 0, 0, 0, 1, 4, 20, 86, 409, 1852, 5440, 4840, 496, 0

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
T(n,k)=Xi_k(P_n) which is the number of non-equivalent distinguishing partitions of the path on n vertices, with at most k parts. Two partitions P1 and P2 of a graph G are said to be equivalent if there is a nontrivial automorphism of G which maps P1 onto P2. A distinguishing partition is a partition of the vertex set of G such that no nontrivial automorphism of G can preserve it. - Bahman Ahmadi, Sep 02 2019

			Array begins with T(1,1):
0   0     0      0       0       0       0       0       0       0 ...
0   0     0      0       0       0       0       0       0       0 ...
0   1     1      1       1       1       1       1       1       1 ...
0   2     4      4       4       4       4       4       4       4 ...
0   6    16     20      20      20      20      20      20      20 ...
0  12    52     80      86      86      86      86      86      86 ...
0  28   169    336     400     409     409     409     409     409 ...
0  56   520   1344    1852    1976    1988    1988    1988    1988 ...
0 120  1600   5440    8868   10168   10388   10404   10404   10404 ...
0 240  4840  21760   42892   54208   57108   57468   57488   57488 ...
0 496 14641  87296  210346  299859  331705  337595  338155  338180 ...
0 992 44044 349184 1038034 1699012 2012202 2091458 2102518 2103348 ...
For T(4,2)=2, the chiral pairs are AAAB-ABBB and AABA-ABAA.
For T(4,3)=4, the above, AABC-ABCC, and ABAC-ABCB.

B. Ahmadi, F. Alinaghipour and M. H. Shekarriz, Number of Distinguishing Colorings and Partitions, arXiv:1910.12102 [math.CO], 2019.

Columns 1-6 are A000004, A122746(n-3), A107767(n-1), A320934, A320935, A320936.
As k increases, columns converge to A320937.
Cf. transpose of A278984 (oriented), A320750 (unoriented), A305749 (achiral).
Partial column sums of A320525.

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
Table[Sum[StirlingS2[n,j] - Ach[n,j], {j,k-n+1}]/2, {k,15}, {n,k}] // Flatten

T(n,k) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where S2 is the Stirling subset number A008277 and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
T(n,k) = (A278984(k,n) - A305749(n,k)) / 2 = A278984(k,n) - A320750(n,k) = A320750(n,k) - A305749(n,k).
T(n,k) = Sum_{j=1..k} A320525(n,j).

A107767 a(n) = (1 + 3^n - 23^(n/2))/4 if n is even, (1 + 3^n - 43^((n-1)/2))/4 if n odd.

Original entry on oeis.org

0, 1, 4, 16, 52, 169, 520, 1600, 4840, 14641, 44044, 132496, 397852, 1194649, 3585040, 10758400, 32278480, 96845281, 290545684, 871666576, 2615029252, 7845176329, 23535617560, 70607118400, 211821620920, 635465659921

Offset: 1

Emeric Deutsch, Jun 12 2005

a(n-1) is the number of chiral pairs of color patterns (set partitions) for a row of length n using up to 3 colors (subsets). For n=4, a(n-1)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB. - Robert A. Russell, Oct 28 2018

Balaban, A. T., Brunvoll, J., Cyvin, B. N., & Cyvin, S. J. (1988). Enumeration of branched catacondensed benzenoid hydrocarbons and their numbers of Kekulé structures. Tetrahedron, 44(1), 221-228. See Eq. 5.

Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Gy. Tasi and F. Mizukami, Quantum algebraic-combinatoric study of the conformational properties of n-alkanes, J. Math. Chemistry, 25, 1999, 55-64 (see p. 60).
Index entries for linear recurrences with constant coefficients, signature (4,0,-12,9).

Cf. A167993 (first differences).
Column 3 of A320751, offset by 1.
Cf. A124302 (oriented), A001998 (unoriented), A182522 (achiral), varying offsets.

a:=[];; for n in [1..30] do if n mod 2 <> 0 then Add(a,(1+3^n-4*3^((n-1)/2))/4); else Add(a,(1+3^n-2*3^(n/2))/4); fi; od; a; # Muniru A Asiru, Oct 30 2018

I:=[0, 1, 4, 16]; [n le 4 select I[n] else 4*Self(n-1)-12*Self(n-3)+9*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jun 26 2012

a:=proc(n) if n mod 2 = 0 then (1+3^n-2*3^(n/2))/4 else (1+3^n-4*3^((n-1)/2))/4 fi end: seq(a(n),n=1..32);

CoefficientList[Series[-x/((x-1)*(3*x-1)*(3*x^2-1)),{x,0,40}],x] (* or *) LinearRecurrence[{4,0,-12,9},{0,1,4,16},50] (* Vincenzo Librandi, Jun 26 2012 *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=3; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,2,40}] (* Robert A. Russell, Oct 28 2018 *)
CoefficientList[Series[(1/12 E^(-Sqrt[3] x) (-3 + 2 Sqrt[3] - (3 + 2 Sqrt[3]) E^(2 Sqrt[3] x) + 3 E^((3 + Sqrt[3]) x) + 3 E^(x + Sqrt[3] x)))/x, {x, 0, 20}], x]*Table[(k+1)!, {k, 0, 20}] (* Stefano Spezia, Oct 29 2018 *)

x='x+O('x^50); concat(0, Vec(x^2/((1-x)*(3*x-1)*(3*x^2-1)))) \\ Altug Alkan, Sep 23 2018

G.f.: -x^2 / ( (x-1)*(3*x-1)*(3*x^2-1) ). - R. J. Mathar, Dec 16 2010
a(n) = 4*a(n-1) - 12*a(n-3) + 9*a(n-4). - Vincenzo Librandi, Jun 26 2012
From Robert A. Russell, Oct 28 2018: (Start)
a(n-1) = Sum_{j=0..k} (S2(n,j) - Ach(n,j)) / 2, where k=3 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(n-1) = (A124302(n) - A182522(n))/2.
a(n-1) = A124302(n) - A001998(n-1).
a(n-1) = A001998(n-1) - A182522(n).
a(n-1) = A122746(n-2) + A320526(n). (End)
E.g.f.: (1/12)*exp(-sqrt(3)*x)*(-3 + 2*sqrt(3) - (3 + 2*sqrt(3))*exp(2*sqrt(3)*x) + 3*exp((3 + sqrt(3))*x) + 3*exp(x + sqrt(3)*x)). - Stefano Spezia, Oct 29 2018
From Bruno Berselli, Oct 31 2018: (Start)
a(n) = (1 + 3^n - 3^((n-1)/2)*(4 + (-2 + sqrt(3))*(1 + (-1)^n)))/4. Therefore:
a(2*k)   = (3^k - 1)^2/4;
a(2*k+1) = (3^k - 1)*(3^(k+1) - 1)/4. (End)

Entry revised by N. J. A. Sloane, Jul 29 2011

A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859, 1699012, 9808848, 57335124, 338073107, 2004955824, 11936998016, 71253827696, 426061036747, 2550545918300, 15280090686256, 91588065861292, 549159350303235, 3293482358956552, 19755007003402944

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244, -4176,11664,-5184).

Column 6 of A320751.

Cf. A056273 (oriented), A056325 (unoriented), A305752 (achiral).

Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=6; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859}, 40]

concat([0,0], Vec(x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^40))) \\ Colin Barker, Nov 22 2018

a(n) = (A056273(n) - A305752(n))/2.
a(n) = A056273(n) - A056325(n).
a(n) = A056325(n) - A305752(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n) + A320529(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=6 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
From Colin Barker, Nov 22 2018: (Start)
G.f.: x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)

A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300).

Column 5 of A320751.

Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=5; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]

a(n) = (A056272(n) - A305751(n))/2.
a(n) = A056272(n) - A056324(n).
a(n) = A056324(n) - A305751(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018

A107659 a(n) = Sum_{k=0..n} 2^max(k, n-k).

Original entry on oeis.org

1, 4, 10, 24, 52, 112, 232, 480, 976, 1984, 4000, 8064, 16192, 32512, 65152, 130560, 261376, 523264, 1047040, 2095104, 4191232, 8384512, 16771072, 33546240, 67096576, 134201344, 268410880, 536838144, 1073692672, 2147418112

Offset: 0

Mitch Harris

Define an infinite array by m(n,k) = 2^n-n+k for n>=k>=0 (in the lower left triangle) and by m(n,k) = 2^k+k-n for k>=n>=0 (in the upper right triangle). The antidiagonal sums of this array are a(n) = sum_{k=0..n} m(n-k,k). - J. M. Bergot, Aug 16 2013

			G.f. = 1 + 4*x + 10*x^2 + 24*x^3 + 52*x^4 + 112*x^5 + 232*x^6 + 480*x^7 + ... - _Michael Somos_, Jun 24 2018

Index entries for linear recurrences with constant coefficients, signature (2,2,-4).

Cf. A027383, A052955.

Table[Sum[2^Max[k,n-k],{k,0,n}],{n,0,30}] (* or *) LinearRecurrence[ {2,2,-4},{1,4,10},30] (* Harvey P. Dale, Nov 10 2013 *)
a[ n_] := 2^(n + 2) - (2 + Mod[n + 1, 2]) 2^Quotient[n + 1, 2]; (* Michael Somos, Jun 24 2018 *)

{a(n) = 2^(n+2) - (2 + (n+1)%2) * 2^((n+1)\2)}; /* Michael Somos, Jun 24 2018 */

a(2n) = 2^n(2^(n+2)-3), a(2n+1) = 2^n(2^(n+3)-4).
G.f.: (1+2*x)/[(1-2*x)*(1-2*x^2)].
a(n) = A122746(n) +2*A122746(n-1). - R. J. Mathar, Aug 16 2013
a(0)=1, a(1)=4, a(2)=10, a(n)=2*a(n-1)+2*a(n-2)-4*a(n-3). - Harvey P. Dale, Nov 10 2013
a(n) = 2^(n+2) - (2 + mod(n+1, 2)) * 2^floor((n+1)/2). - Michael Somos, Jun 24 2018
a(n) = - (2^(n+2)) * A052955(-n-3) for all n in Z. - Michael Somos, Jun 24 2018

A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Column 4 of A320751.

Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral).

Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 40] (* or *)
Ach[n_, k_] := Ach[n, k] = If[n<2, Boole[n==k && n>=0], k Ach[n-2,k] + Ach[n-2,k-1] + Ach[n-2,k-2]] (* A304972 *)
k=4; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

a(n) = (A124303(n) - A305750(n))/2.
a(n) = A124303(n) - A056323(n).
a(n) = A056323(n) - A305750(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).
a(2*m) = (16^m - 4*4^m)/48.
a(2*m-1) = (16^m - 4*4^m)/192.
a(n) = (4^n - 4^floor(n/2+1))/48.
G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018
a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018

A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 5, 5, 1, 1, 8, 10, 8, 1, 1, 9, 18, 18, 9, 1, 1, 12, 27, 40, 27, 12, 1, 1, 13, 39, 67, 67, 39, 13, 1, 1, 16, 52, 112, 134, 112, 52, 16, 1, 1, 17, 68, 164, 246, 246, 164, 68, 17, 1, 1, 20, 85, 240, 410, 504, 410, 240, 85, 20, 1

Offset: 0

Gary W. Adamson, Jan 01 2008

			First few rows of the triangle are:
  1;
  1,   1;
  1,   4,   1;
  1,   5,   5,   1;
  1,   8,  10,   8,   1;
  1,   9,  18,  18,   9,   1;
  1,  12,  27,  40,  27,  12,   1;
  1,  13,  39,  67,  67,  39,  13,   1;
  1,  16,  52, 112, 134, 112,  52,  16,   1;
  1,  17,  68, 164, 246, 246, 164,  68,  17,   1;
  ...

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A034851, A042948, A077957, A122746 (row sums).

A136489:= func< n,k | 2*Binomial(n,k) - Binomial(n mod 2, k mod 2)*Binomial(Floor(n/2), Floor(k/2)) >;
[A136489(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Aug 01 2023

T[n_, k_]:= 2*Binomial[n,k] -Binomial[Mod[n,2], Mod[k,2]]*Binomial[Floor[n/2], Floor[k/2]];
Table[T[n,k], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Aug 01 2023 *)

def A136489(n,k): return 2*binomial(n,k) - binomial(n%2, k%2)*binomial(n//2, k//2)
flatten([[A136489(n,k) for k in range(n+1)] for n in range(13)]) # G. C. Greubel, Aug 01 2023

T(n, k) = 3*A007318(n, k) - 2*A034851(n, k).
Sum_{k=0..n} T(n, k) = A122746(n).
From G. C. Greubel, Aug 01 2023: (Start)
T(n, k) = 2*A007318(n, k) - A051159(n, k).
T(n, k) = T(n-1, k) + T(n-1, k-1) if k is even.
T(n, n-k) = T(n, k).
T(n, n-1) = A042948(n).
Sum_{k=0..n} (-1)^k * T(n, k) = 2*[n=0] - A077957(n). (End)

A156665 Triangle read by rows, A156663 * A007318.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 3, 5, 3, 1, 7, 8, 8, 4, 1, 7, 15, 16, 12, 5, 1, 15, 22, 31, 28, 17, 6, 1, 15, 37, 53, 59, 45, 23, 7, 1, 31, 52, 90, 112, 104, 68, 30, 8, 1, 31, 83, 142, 202, 216, 172, 98, 38, 9, 1, 63, 114, 225, 344, 418, 388, 270, 136, 47, 10, 1

Offset: 0

Gary W. Adamson, Feb 12 2009

Row sums = A122746: (1, 2, 6, 12, 28, 56, 120,...).

			First few rows of the triangle =
1;
1, 1;
3, 2, 1;
3, 5, 3, 1;
7, 8, 8, 4, 1;
7, 15, 16, 12, 5, 1;
15, 22, 31, 28, 17, 6, 1;
15, 37, 53, 59, 45, 23, 7, 1;
31, 52, 90, 112, 104, 68, 30, 8, 1;
31, 83, 142, 202, 216, 172, 98, 38, 9, 1;
63, 114, 225, 344, 418, 388, 270, 136, 47, 10, 1;
...

Robert Israel, Table of n, a(n) for n = 0..10010 (first 141 rows, flattened)

A156663, A122746

N:= 12: # for the first N rows
A156663:= Matrix(N,N,(i,j) -> `if`((i-j)::even, 2^((i-j)/2),0), shape=triangular[lower]):
A007318:= Matrix(N,N,(i,j) -> binomial(i-1,j-1),shape=triangular[lower]):
P:= A156663 . A007318:
seq(seq(P[i,j],j=1..i),i=1..N); # Robert Israel, Aug 10 2015

Triangle read by rows, A156663 * A007318

G.f. for triangle: 1/((1-2*x^2)*(1-x-x*y)). - Robert Israel, Aug 10 2015

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A136489 Triangle T(n, k) = 3A007318(n, k) - 2A034851(n, k).